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2 years ago

14

To drive safely, the air pressure in your tires should be between 20 and 30 psi. Write this as an absolute value inequality, wit

h x as the air pressure.

1 answer:

Westkost [7]2 years ago

7 0

Given that for a safe drive the pressure of the car should be between 20 and 30 psi. This implies that the pressure in the tires should strictly be between 20 and 30, hence given that the x is the pressure value then the absolute value inequality will be:
20≤ΙxΙ≤30

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Now, $\sqrt{x^2}=|x|$ in general. But since we want our substitution $x=5\sin\theta$ to be invertible, we are tacitly assuming that we're working over a restricted domain. In particular, this means $\theta=\sin^{-1}\dfrac x5$ , which implies that $\left|\dfrac x5\right|\le1$ , or equivalently that $|\theta|\le\dfrac\pi2$ . Over this domain, $\cos\theta\ge0$ , so $\sqrt{\cos^2\theta}=|\cos\theta|=\cos\theta$ .

Long story short, this allows us to go from

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$\displaystyle5\int\cos\theta\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
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$\displaystyle5\left(\int\csc\theta\,\mathrm d\theta-\int\sin\theta\,\mathrm d\theta\right)$
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and using basic trigonometric properties (e.g. Pythagorean theorem) this reduces to

$=-5\ln\left|\dfrac{5+\sqrt{25-x^2}}x\right|+\sqrt{25-x^2}+C$

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